Poincaré-Birkhoff-Witt theorems in higher algebra.

We extend the classical Poincaré-Birkhoff-Witt (PBW) theorem to higher algebra by establishing a version that applies to spectral Lie algebras. We deduce this statement from a basic relation between operads in spectra: the commutative operad is the quotient of the associative operad by a right action of the spectral Lie operad. This statement, in turn, is a consequence of a fundamental relation between different En-operads, which we articulate and prove. We deduce a variant of the Poincaré-Birkhoff-Witt theorem for relative enveloping algebras of En-algebras. Our methods also give a simple construction and description of the higher enveloping En-algebras of a spectral Lie algebra. This article is part of the theme issue 'Derived Lie algebras in geometry and topology'.